When a data bit sequence comes to a modulator, it is grouped with a certain number of bits and each group is mapped to one of the points on the constellation as shown in FIG. 1. A point on the constellation is called a symbol and a symbol is a complex number that has phase and magnitude information. FIG. 2 shows one constellation which has eight symbols with one magnitude that is a set distance between the point and the origin. Referring to FIG. 2, each symbol consists of three bits with a different bit sequence. The integer number of each point is a decimal number representing the bit sequence. The gray code is used to assign the bit sequence so that the bit difference between two closest points is always one. The symbols are transmitted over the channel and the position of that received symbol may be changed due to the noise.
When the signal is received at the receiver, the receiver is trying to find out which point on the constellation is the closest to the received signal. This is the demodulation process. The closest one is considered the transmitted one at the receiver. This process is explained in another way using the region. Each point on the constellation has its own region as shown FIG. 3. If the received symbol is fallen in one of regions, the point of that region is considered as the transmitted symbol. Referring to FIG. 3, RX is the received symbol. The closest point on the constellation is 0 and RX has fallen in the region of 0. Therefore, the demodulated signal is 0.
In differential modulation, the symbol is not transmitted with the phase that the point has. FIG. 4 explains a differential modulation process. Referring to FIG. 4, “A” is a mapped current point with incoming bit sequence that will be modulated and R is a differentially modulated symbol with a previous symbol. The phase of A, P2, is added with the phase of R, P1. TX is the modulated A with the added phase, P1+P2 and is transmitted. There is no magnitude change of A while there is phase change. At the receiver, the phase of the received symbol is subtracted by the phase of the previous received one. This is called a differential phase shift keying. Using this scheme the received signal is demodulated at the receiver without knowing the information of the phase change distorted by the noise at the receiver as well as the one of the magnitude distortion. In coherent system, the information of phase shift and channel gain is estimated by a channel estimation process, which makes the system complicated.
The constellation for the conventional Differential Phase Shift Keying (DPSK) has one magnitude and different phase changes. Referring to FIG. 5a to 5b, all points are placed on one ring with the same phase difference between two consecutive points. The performance of this modulation is degraded for constellations with more than 16 points because the signal points are located very closely on one circle. The distance between two consecutive points in FIG. 5b is closer than one in FIG. 5a. 
In order to overcome the deficiency of DPSK, there are different modulation and demodulation for a differential encoding. FIG. 6 shows the constellation and the modulation process of a Differential Amplitude Phase Shift Keying (DAPSK). In this scheme, the magnitude and the phase of the symbol is encoded differentially. However, the constellation of this scheme is not efficient for the symbols positioned in the inner ring since the symbols are too close in such rings.